x, a = 0, b = 1, (c) f(x

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MATHEMATICAL ANALYSIS PROBLEMS LIST 12

4.01.10

(1) Find the formula for Cn = Xn

i=1

b − a n f¡

a + ib − a n

¢, and the compute lim

n→∞Cn: (a) f(x) = 1, a = 5, b = 8, (b) f(x) = x, a = 0, b = 1,

(c) f(x) = x, a = 1, b = 5, (d) f(x) = x2, a = 0, b = 5, (e) f(x) = x3, a = 0, b = 1, (f) f(x) = 2x + 5, a = −3, b = 4, (g) f(x) = x2+ 1, a = −1, b = 2, (h) f(x) = x3+ x, a = 0, b = 4, (i) f(x) = ex, a = 0, b = 1.

(2) Compute the following denite integrals by constructing a sequence of partitions of the interval, corresponding Riemann sums, and their limits:

(a) Z 4

2

x10dx, (xi = 2 · 2i/n), (b) Z e

1

log(x)

x dx, (xi = ei/n), (c)

Z 20

0

x dx, (d)

Z 10

1

e2xdx, (e)

Z 1

0

3

x dx, (xi = ni33), (f) Z 1

−1

|x| dx, (g)

Z 2

1

dx

x dx, (xi = 2i/n), (h) Z 4

0

√x dx, (xi = 4in22).

(3) Compute the denite integrals:

(a) Z π

−π

sin(x2007) dx, (b) Z 2

0

arctan([x]) dx, (c)

Z 2

0

[cos(x2)] dx, (d) Z 1

0

√1 + x dx,

(e)

Z −1

−2

1

(11 + 5x)3dx, (f) Z 2

−13

1 p5

(3 − x)4 dx, (g)

Z 1

0

x

(x2+ 1)2 dx, (h) Z 3

0 sgn (x3− x) dx, (i)

Z 1

0

x e−xdx, (j)

Z π/2

0

x cos(x) dx, (k)

Z e−1

0

log(x + 1) dx, (l) Z π

0

x3 sin(x) dx, (m)

Z 9

4

√x

√x − 1dx, (n)

Z e3

1

1 xp

1 + log(x)dx, (o)

Z 2

1

1

x + x3 dx, (p)

Z 2

0

1

x + 1 +p

(x + 1)3 dx, (q)

Z 5

0

|x2− 5x + 6| dx, (r) Z 1

0

ex

ex− e−xdx,

1

(2)

(s) Z 2

1

x log2(x) dx, (t)

Z 7

0

x3

3

1 + x2 dx, (u)

Z

0

| sin(x)| dx, (w)

Z π/2

0

cos(x) sin11(x) dx, (x)

Z log 5

0

ex ex− 1

ex+ 5 dx, (y) Z π

−π

x2007cos(x) dx, (z)

Z

0

(x − π)2007cos(x) dx. (4) Prove the following estimates:

(a)

Z π/2

0

sin(x)

x dx < 2, (b) 1

5 <

Z 2

1

1

x2+ 1dx < 1 2, (c) 1

11 <

Z 10

9

1

x + sin(x)dx < 1

8, (d) Z 2

−1

|x|

x2+ 1dx < 3 2, (e)

Z 1

0

x(1 − x99+x) dx < 1

2, (f) 2

2 <

Z 4

2

x1/xdx, (g) 5 <

Z 3

1

xxdx < 31, (h) Z 2

1

1

xdx < 3 4. (5) Compute the following limits:

(a) limn→∞¡1

n+ n+11 +n+21 +n+31 + · · · + 2n1 ¢ , (b) lim

n→∞

¡120+220+320+···+n20 n21

¢, (c) lim

n→∞

¡ 1

n2 +(n+1)1 2 +(n+1)1 2 + (n+3)1 2 + · · · +(2n)1 2

¢· n, (d) limn→∞¡ 1

n

2n +n12n+1 + n12n+2 + n12n+3 + · · · + n13n¢ , (e) limn→∞¡

sin(n1) + sin(2n) + sin(3n) + · · · + sin(nn

· 1n, (f) lim

n→∞

¡√4n +√

4n + 1 +√

4n + 2 + · · · +√ 5n¢

· n1n, (g) lim

n→∞

¡ 1

3

n+ 3 1

n+1 + 3 1

n+2+ · · · + 31

8n

¢· 31

n2, (h) lim

n→∞

¡6

n·(3 n+3

n+1+3

n+2+···+3

2n) n+

n+1+

n+2+···+ 2n

¢, (i) limn→∞¡n

n2 +n2n+1 + n2n+4 +n2n+9 +n2+16n + · · · + n2+nn 2

¢, (j) lim

n→∞

¡4

5n+ 5n+34 +5n+64 + 5n+94 + · · · + 26n4 ¢ , (k) lim

n→∞

¡ 1

7n + 7n+21 +7n+41 + 7n+61 + · · · + 9n1 ¢ , (l) limn→∞¡ 1

7n2 + 7n21+1 +7n21+2 +7n21+3 + · · · + 8n12

¢,

(m) lim

n→∞

1 n

¡e√1

n + e√2

n + e√3

n + · · · + e√n

n¢ , (n) limn→∞¡ 1

n+n+31 +n+61 +n+91 + · · · + 17n¢ 1

n, (o) limn→∞¡n2+0

(3n)3 +(3n+1)n2+13 +(3n+2)n2+23 + (3n+3)n2+33 + · · · + n(4n)2+n3

¢, (p) lim

n→∞

¡ n

2n2 +2(n+1)n 2 +2(n+2)n 2 + 2(n+3)n 2 + · · · + 50nn2

¢, (r) lim

n→∞

¡ n

2n2 + n2+(n+1)n 2 + n2+(n+2)n 2 + n2+(n+3)n 2 + · · · + 50nn2

¢.

2

Figure

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